具有多项式系数的二阶线性微分方程解的零点分布

On the Distribution of Zeros of the Solutions of Second Order Linear Differential Equations with Polynomial CoeflicientsCAI HUIHui-jing

本文研究具有多项式系数的二阶线性微分方程解的零点分布,细化了Bank和Laine的结果。证明当n为偶数时,对任意正整数k,总可取系数A(z)为n次多项式,使得方程∫+A(z)f=0存在非平凡解f有k个零点(按重数计)。进一步.我们还给出了该方程存在无零点解的条件。特别地.当系数A(z)=z^(2m)时.我们证明该方程非平凡解的零点序列的收敛级都等于其增长级。

In this paper, we investigate the distribution of zeros of the solutions of second order linear differential equations with polynomial coefficients.The result of Bank and Laine has been improved. It is proved that for any,some solution of just has k zeros, if n is even and A（z） is the constant polynomial of degree n. Moreover,we give a condition such that the equation has a solution having no zeros. Particularly, it is proved that if, the exponent of convergence of the zero - sequence of, which is a nontrivial solution of the equation, ism＋ 1.